Wasn't it Fleetie who wrote:
Forgetting the magnetic pole, then:

At the true North Pole, i.e. that about which the Earth is rotating at a
given time, would a GPS read exactly 90 deg. N?

How would the precession of the Earth affect this?

Precession doesn't affect it. There are in fact three types of
astronomical precession that the Earth undergoes, but none of them
affect the point around which the Earth rotates or affect GPS
satellites. For example, the precession of the equinoxes changes the
point in space towards which the rotation axis is pointing, but the axis
retains its position relative to the Earth. Like a tumbling gyroscope,
the axis moves in space, but the gyroscope still spins around that axis.

A totally separate geological effect is polar wobble. The axis of
rotation wobbles by a few metres.

GPS systems operate natively in the WGS84 reference frame. That uses a
fixed point for its North Pole, rather than following the polar wobble.
That has the advantage that my house retains exactly the same lat/lng
values in the WGS84 reference frame. In earlier systems that used the
rotational axis as the reference datum, it was discovered that precision
measurements exhibited measurable variations. Such effects could be
rather awkward if you lived in a country that has a boundary defined by
a particular line of latitude.

When an accurate GPS reads exactly 90 deg, it's within a few metres of
the rotation axis.

--
Mike Williams
Gentleman of Leisure